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Theorem islhp2 34095
Description: The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 18-May-2012.)
Hypotheses
Ref Expression
lhpset.b 𝐵 = (Base‘𝐾)
lhpset.u 1 = (1.‘𝐾)
lhpset.c 𝐶 = ( ⋖ ‘𝐾)
lhpset.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
islhp2 ((𝐾𝐴𝑊𝐵) → (𝑊𝐻𝑊𝐶 1 ))

Proof of Theorem islhp2
StepHypRef Expression
1 lhpset.b . . 3 𝐵 = (Base‘𝐾)
2 lhpset.u . . 3 1 = (1.‘𝐾)
3 lhpset.c . . 3 𝐶 = ( ⋖ ‘𝐾)
4 lhpset.h . . 3 𝐻 = (LHyp‘𝐾)
51, 2, 3, 4islhp 34094 . 2 (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊𝐵𝑊𝐶 1 )))
65baibd 946 1 ((𝐾𝐴𝑊𝐵) → (𝑊𝐻𝑊𝐶 1 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977   class class class wbr 4578  cfv 5790  Basecbs 15644  1.cp1 16810  ccvr 33361  LHypclh 34082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-lhyp 34086
This theorem is referenced by:  lhpoc  34112
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